Ratio & Proportion

Spec reference: Ratio, proportion and rates of change
Key idea: Simplify ratios, divide quantities in a given ratio, and solve proportion problems.

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Simplifying ratios

Divide all parts of the ratio by their HCF.

Example

Simplify $15:25$

HCF of 15 and 25 is 5:

$$15:25=\boxed{{\displaystyle 3:5}}$$

Example

Simplify $1.2:3$

Multiply both by 10 to clear the decimal, then simplify:

$$12:30=\boxed{{\displaystyle 2:5}}$$

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Dividing in a given ratio

1. Find the total number of parts
2. Find the value of one part
3. Multiply for each share

Example

Share £240 in the ratio $3:5$

Total parts: $3+5=8$

One part: $240÷8=30$

$$\text{First share}=3\times 30=\pounds 90$$$$\text{Second share}=5\times 30=\pounds 150$$

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Using ratios to find amounts

Example

In a class the ratio of boys to girls is $2:3$. There are 18 girls. How many boys are there?

3 parts = 18, so 1 part = $18÷3=6$

Boys = $2\times 6=\boxed{{\displaystyle 12}}$

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Direct proportion

Two quantities are in direct proportion if, when one doubles, the other doubles.

$$y=kx$$

Example

5 pens cost £3.50. Find the cost of 8 pens.

Cost of 1 pen: $3.50÷5=0.70$

Cost of 8 pens: $0.70\times 8=\boxed{{\displaystyle \pounds 5.60}}$

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Inverse proportion

Two quantities are in inverse proportion if, when one doubles, the other halves.

$$y=\frac{k}{x}$$

Example

4 workers take 15 days to complete a job. How long would 6 workers take?

Total work: $4\times 15=60$ worker-days

Time for 6 workers: $60÷6=\boxed{{\displaystyle 10\text{ days}}}$

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Best value problems

Divide the cost by the quantity to find the cost per unit.

Example

Which is better value?
Tin A: 400g for £1.20
Tin B: 650g for £1.95

Tin A: $\pounds 1.20÷400=0.3\text{p per gram}$

Tin B: $\pounds 1.95÷650=0.3\text{p per gram}$

They are equal value.

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Exam tips

Watch out for

• Always simplify ratios fully
• Check your answers add up to the original total when dividing in a ratio
• For inverse proportion: more workers = fewer days (they go in opposite directions)

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Test Yourself

Question 1 of 5

Simplify the ratio 24 : 36